Gina's link summed up Galveston pretty well, so I'll just add Houston stuff. You might want to look into buying a Houston city pass.
http://www.citypass.com/houston With the city pass you get:
1) Admission to the Space Center (a must do when you visit Houston)
2) Admission to the Downtown Aquarium (not a must-see, but if it's included, definitely worth the trip). The restaurant there has a huge aquarium in it, which I've heard is a lot like Coral Reef at Epcot (I've never been to coral reef, though, so I can't say for sure).
3) Admission to the Museum of Natural Science (they have a fabulous new dinosaur exhibit, as well as a new permanent Egyptian Exhibit)
4) Admission to the Zoo OR a ride pass to Kemah's Boardwalk If you're into zoos, the zoo is great-- its Africa area was actually created by someone who used to work for Disney. But, with your age kiddo's Kemah might go over better.
5)Admission to the Children's Museum OR to the Museum of Fine Arts You guys have outgrown the children's museum most likely (although it is awesome). The Fine Arts museum has a special exhibit right now by an artist who works with lights. I've heard it's really cool, and I'm pretty sure it's included in regular admission, although they advise to make a reservation to see it, and I'm not sure how one would do that with a city pass. Here's a link to the exhibit, though:
http://www.mfah.org/exhibitions/james-turrell-retrospective/
The city pass is $46 for adults and $36 for kids. You can have a week's worth of fun doing all that's included in the pass for less than a day at Disney

! Have fun here

. Just be warned, it is HOT, although it's been raining a ton recently, so it hasn't been ungodly hot for the past few days. --Kaite